Given 2 states with energy , and interaction energy between the two states , the Hamiltonian of the system is

This system is already discussed in this post. The solution state in here again as:

The eigen-energies

the eigen-states are

where

The are the wavefunction of pure state of .

This can be easily extended to 3-state system or -state system. The number of coupling constants is .

Experimentally, we observed the mixed states and the spectroscopic factors originated from one of the pure state. For example, in a neutron transfer reaction, the transferred neutron may coupled with the 2+ state of the core. The neutron is sitting in a pure orbital (), say 1d5/2 orbital, it couple with the 2+ state () with interaction energy . The final states will have energy and we extract the spectroscopic factors () for the 1d5/2 orbital $.

To state more clear, in a transfer reaction, we may observed two excited states with transfer of neutron. This neutron assumed to couple with a core state and a core excited state . And we supposed that the 2 observed states are:

We then want to find out the un-perturbed energy and the interaction . And this is actually an easy problem by the diagonalization process.

The Hamiltonian was diagonalized into

where

Thus,